We next input thus determined atomic displacement fields into the formulas for the continuum Hamiltonian developed in the previous paper, expanding up to second order in gradients in the intra-layer Hamiltonian and up to first order in gradients in the inter-layer Hamiltonian.
For the intra-layer Hamiltonian, \(H_{intra}\) , we find an efficient way to compute the desired parameters of the continuum model from the microscopic tight binding functions of Ref. [1]} and Ref. [2]} by Poisson resumming the powerlaw decaying momentum space sums into real space where they fall off exponentially fast. Each moment of the position vector weighted with the intra-layer hopping function is accompanied by a gradient of either the fermion field or the atomic displacement field, and contributes a factor of \(|\mathbf {g}|a\) , where \(|\mathbf {g}|\sim |K|\theta \)  (with \(\theta \)  in radians), i.e. a factor of \(\sim 0.08\) .
For the inter-layer Hamiltonian \(H_{inter}\) , the tunneling falls off fast in the momentum space once the wavevector significantly exceeds the inverse of the inter-layer separation \(1/d_0\) , making the direct momentum summation efficient. At the same time, each moment of the position vector weighted with the inter-layer hopping function is also accompanied by a gradient of either the fermion field or the atomic displacement field, and thus contributes a factor of \(\sim |\mathbf {g}|d_0\) . Since \(d_0\simeq 1.36a\) , the higher order gradient terms are suppressed by similar factors in the \(H_{intra}\)  and \(H_{inter}\) .
